Baroque conjectures

Let me address the views (guesses) on the four questions, about the perfect and baroque numbers, listed in the previous post:

• everybody believes that there are infinitely many (even) perfect numbers, because everybody believes that there are infinitely many Mersenne prime numbers p (i.e. prime numbers p such that 2^p-1 is a prime too);

• it seems to me that some (many? most?) specialists believe that there are only finitely many baroque numbers which are not perfect; some even think that all baroque non-perfect numbers are already known (no way!). Myself, I believe that there are infinitely many of them but finitely many for each baroqueness coefficient;

• nobody believes that there is any odd perfect number;

• I think that nobody believes that there is any odd baroque number; I don't either but I am not so sure :-).

I am interested in the race toward finding more baroque numbers.

Back to baroque numbers?

The classical, standard name for my term baroque numbers is pluperfect or polyperfect
numbers or similar. A natural number n (n = 1 2 ...) is called baroque when the sum sd(n) of all natural divisors of n is divisible by n; let's call

brq(n) := sd(n)/n

the baroqueness of n. When a baroqueness of a number is two, brq(n) = 2, then it is called perfect. The smallest perfect number, known since the ancient times, is n=6. Indeed, in this case

sd(n) = 1+2+3+4+6 = 12 = 2*n

Questions about the baroque numbers are among the oldest, open (unresolved) problems of the whole mathematics. They are difficult and, today, somewhat isolated from the main developments, hence they are not as popular among the best mathematicians as they used to be in the past, when Euclid, Fermat, Decart, Euler and others were interested in them. Let me list the main open questions:

• are there infinitely many perfect numbers?
• are there infinitely many baroque numbers?
• does there exist an odd perfect number?
• does there exist an odd baroque number > 1?

Of course n=1 is the only number for which brq(n) = 1.

The Euclid-Euler tandem (:-) proved that an even natural number n is perfect if and only if there exists a (unique) natural number p such that the following two conditions hold:

• 2^p - 1 is prime;
• n = 2^p-1*(2^p-1)

It is well known and easy to see that when 2^p-1 is prime then so must be the exponent p itself.

Another old result (an ancient observation) is that brq(120) = 3, i.e. 120 is a baroque number, and its baroqueness is 3.